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Map from Wilson loop into the Riemann surface Zeros determine shape of the WL. z can be written as:

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Conclusions We argue that there is an infinite parameter family of closed Wilson loops whose dual surfaces can be found analytically. The world-sheet has the topology of a disk and the renormalized area is found as a finite one dimensional contour integral over the world- sheet boundary. We showed specific examples for g=3. Integrability properties of Euclidean Wilson loops deserve further study.